6-Dimensional Analysis

To convert 7.580 micrograms to megagrams, we convert to the base unit (grams) first. We can define conversion factors by making the larger prefix 'one' (e.g., 1 gram = 10^6 micrograms) or the smaller prefix 'one' (e.g., 1 microgram = 10^-6 grams). After setting up and canceling units, the calculation yields 7.580 x 10^-12 megagrams, retaining four significant figures.

Dimensional analysis, also known as the factor-label method in math, is a method to convert from one unit to another using conversion factors. Conversion factors are equivalent statements, such as one yard being equal to three feet. The method involves starting with the given value, identifying conversion factors, ensuring diagonal units cancel, and calculating with correct significant digits.

To convert 530 seconds to days, we identify conversion factors: 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day. We start with 530 seconds, cancel units diagonally (seconds-minutes-hours-days), multiply all numerators, divide by all denominators, and apply significant figures. The result is 0.0061 days after rounding to two significant figures.

To convert 4.08 micrometers to centimeters, we go through the base unit, meters. We use the conversion factors: 1 meter = 10^6 micrometers and 1 meter = 100 centimeters. After setting up the conversion and canceling units, the calculation results in 4.08 x 10^-4 centimeters, maintaining three significant figures.

To convert 99.30 kiloliters to cubic centimeters, we use the key conversion: 1 milliliter = 1 cubic centimeter. We first convert kiloliters to milliliters (kiloliters to liters, then liters to milliliters). The conversion factors are 1 kiloliter = 1000 liters and 1 liter = 1000 milliliters. The final answer is 9.930 x 10^7 cubic centimeters, with four significant figures.

To convert 1.00 kilometers cubed to centimeters cubed, we convert linear units and then cube the entire conversion factor. The conversion factors are 1 kilometer = 1000 meters and 1 meter = 100 centimeters. Each conversion factor is cubed to match the units. The calculation produces 1.000 x 10^15 centimeters cubed, preserving four significant figures.

To convert 0.000000000340 hectoseconds squared to megaseconds squared, we use the base unit of seconds. The conversion factors are 1 hectosecond = 100 seconds and 1 megasecond = 10^6 seconds. These conversion factors are squared in the calculation to cancel out the squared units. The final answer is 3.40 x 10^-11 megaseconds squared, maintaining three significant figures.

To convert 0.0100 kilometers per hour to millimeters per second, we convert numerator units (kilometers to millimeters) and denominator units (hours to seconds) separately. Conversion factors are 1 kilometer=1000 meters, 1 meter=1000 millimeters, 1 hour=60 minutes, and 1 minute=60 seconds. The calculation yields 2.78 millimeters per second, rounded to three significant figures.